# Orthogonal Matrices - Examples with Solutions

## Definition of Orthogonal Matrices

An n × n matrix whose columns form an orthonormal set is called an orthogonal matrix.
As a reminder, a set of vectors is orthonormal if each vector is a unit vector ( length or norm of the vector is equal to n × n and each vector in the set is orthogonal to all other vectors in the set.
These are examples of orthogonal matrices.

## Properties of Orthogonal Matrices

A list of the most important properties of orthogonal matrices is given below. If Q is an orthogonal matrix, then

1.    Q-1 = QT ; this is the most important property of orthogonal matrices as the inverse is simply the transpose.
2.   the rows of Q form an orthonormal set.
3.    Q-1 is an orthogonal matrix
4.   Det( Q ) = ± 1
5.   if ? is an eigrnvalue of ( Q ) , then | ? | = 1
6.   if Q1 and Q2 are n × n orthogonal matrices, then Q1 Q2 is also an orthogonal matrix.

## Examples with Solutions

 

Example 1
The matrices $$Q_1 = \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}$$ and $$Q_2 = \begin{bmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & -1 \end{bmatrix}$$ are orthogonal. Verify that the product $$Q_1 Q_2$$ is also orthogonal (Property 6 above)

Solution
We first calculate the product $$Q_1 Q_2$$
$$Q_1 Q_2 = \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & -1 \end{bmatrix} = \begin{bmatrix} 0&0&-1\\ 0&-1&0\\ 1&0&0 \end{bmatrix}$$

Let $$\textbf v_1 , \textbf v_2 , \textbf v_3$$ be the columns of the matrix $$Q_1 Q_2$$ found above.
$$\textbf {v}_1 = \begin{bmatrix} 0\\ 0\\ 1 \end{bmatrix}$$ , $$\textbf {v}_2 = \begin{bmatrix} 0\\ -1\\ 0 \end{bmatrix}$$ , $$\textbf {v}_3 = \begin{bmatrix} -1\\ 0\\ 0 \end{bmatrix}$$

Let us calculate the length or norm of each column
$$|| \textbf {v}_1 || = \sqrt {0^2+0^2+1^2} = 1$$
$$|| \textbf {v}_2 || = \sqrt {0^2+(-1)^2+0^2} = 1$$
$$|| \textbf {v}_3 || = \sqrt {(-1)^2+0^2+0^2} = 1$$

All three vectors are unit vectors.

Calculate the inner product of all pairs of vectors that can be made from the vectors $$\textbf v_1 , \textbf v_2 , \textbf v_3$$
$$\textbf {v}_1 \cdot \textbf {v}_2 = 0 \cdot 0 + 0 \cdot (-1) + 1 \cdot 0 = 0$$
$$\textbf {v}_1 \cdot \textbf {v}_3 = 0 \cdot (-1) + 0 \cdot 0 + 1 \cdot 0 = 0$$
$$\textbf {v}_2 \cdot \textbf {v}_3 = 0 \cdot (-1) + (-1) \cdot 0 + 0 \cdot 0 = 0$$
The three vectors form an orthogonal set.
The three columns of the matrix $$Q_1 Q_2$$ are orthogonal and have norm or length equal to 1 and are therefore orthonormal.

Example 2
Use a calculator to find the inverse of the orthogonal matrix $$Q = \begin{bmatrix} 0 & 0 & 1 \\ -1 & 0 & 0 \\ 0 & -1 & 0 \end{bmatrix}$$ and verify Property 1 above.

Solution
Use any matrix calculator to find
$$Q^{-1} = \begin{bmatrix} 0 & -1 & 0 \\ 0 & 0 & -1 \\ 1 & 0 & 0 \end{bmatrix}$$

Find the transpose of matrix $$Q$$
$$Q^T = \begin{bmatrix} 0 & -1 & 0 \\ 0 & 0 & -1 \\ 1 & 0 & 0 \end{bmatrix}$$
Hence $$Q^{-1} = Q^T$$ , property 1 above.

Example 3
Find the real constants $$a$$ and $$b$$ in the matrix $$Q = \begin{bmatrix} -1 & 0 & 0 \\ 0 & \dfrac{1}{\sqrt 2} & a \\ 0 & \dfrac{1}{\sqrt 2} & b \end{bmatrix}$$ such that $$Q$$ is orthogonal.

Solution
Let $$\textbf v_1 , \textbf v_2 , \textbf v_3$$ be the columns of the matrix $$Q$$ given above such that
$$\textbf {v}_1 = \begin{bmatrix} -1\\ 0\\ 0 \end{bmatrix}$$ , $$\textbf {v}_2 = \begin{bmatrix} 0\\ \dfrac{1}{\sqrt 2}\\ \dfrac{1}{\sqrt 2} \end{bmatrix}$$ , $$\textbf {v}_3 = \begin{bmatrix} 0\\ a\\ b \end{bmatrix}$$

Two sets of conditions for matrix $$Q$$ to be orthogonal:
1) The norm of each column $$\textbf v_1 , \textbf v_2 , \textbf v_3$$ must equal to 1
$$|| \textbf v_1 || = \sqrt {(-1)^2+0^2+0^2} = 1$$
$$|| \textbf v_2 || = \sqrt {0^2+ \left(\dfrac{1}{\sqrt 2} \right)^2+\left(\dfrac{1}{\sqrt 2} \right)^2 } = 1$$
$$|| \textbf v_3 || = \sqrt {0^2+a^2+b^2} = \sqrt {a^2+b^2} = 1$$         (I)

2) The inner product of any two vectors must be equal to zero (orthogonal vectors)
$$\textbf {v}_1 \cdot \textbf {v}_2 = (-1) \cdot 0 + 0 \cdot \left(\dfrac{1}{\sqrt 2} \right) + 0 \cdot \left(\dfrac{1}{\sqrt 2} \right) = 0$$
$$\textbf {v}_1 \cdot \textbf {v}_3 = (-1) \cdot 0 + 0 \cdot a + 0 \cdot b = 0$$
$$\textbf {v}_2 \cdot \textbf {v}_3 = 0 \cdot 0 + \dfrac{1}{\sqrt 2} \cdot a + \dfrac{1}{\sqrt 2} \cdot b = \dfrac{1}{\sqrt 2} \cdot a + \dfrac{1}{\sqrt 2} \cdot b = 0$$         (II)

For all conditions to be satisfied, we need to solve equations (I) and (II) above
Equation (II) above gives $$a = - b$$

Substitute $$a$$ by $$- b$$ in equation (I) to obtain
$$\sqrt {2 b^2 } = 1$$

Solve to obtain
$$b = \pm \dfrac{1}{\sqrt 2}$$

Two solutions to the above question
$$a = - \dfrac{1}{\sqrt 2}$$ and $$b = \dfrac{1}{\sqrt 2}$$
$$a = \dfrac{1}{\sqrt 2}$$ and $$b = - \dfrac{1}{\sqrt 2}$$

## Questions (with solutions given below)

• Part 1
Matrix $$Q = \begin{bmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$ is orthogonal. Verify the first 5 properties, listed above, for matrix $$Q$$.
• Part 2
1) Which of the following matrices are orthogonal?
a) $$A = \begin{bmatrix} \dfrac{1}{\sqrt 2} & \dfrac{1}{\sqrt 2} \\ - \dfrac{1}{\sqrt 2} & \dfrac{1}{\sqrt 2} \end{bmatrix}$$ , b) $$B = \begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{bmatrix}$$ , c) $$C = \begin{bmatrix} \dfrac{1}{\sqrt {10}} & - \dfrac{3}{\sqrt {10}} & 0 \\ 0 & 0 & 1 \\ \dfrac{3}{\sqrt {10}} & \dfrac{1}{\sqrt {10}} & 0 \end{bmatrix}$$
• Part 3
1) Find all matrices of the form $$A = \begin{bmatrix} p & q\\ \dfrac{1}{\sqrt {3}} & r \end{bmatrix}$$ that are orthogonal.
• Part 4
Find the inverse of matrix $$A = \begin{bmatrix} 0 & \dfrac{5}{3 \sqrt 5} & \dfrac{2}{3} \\ \dfrac{1}{\sqrt 5} & - \dfrac{4}{3 \sqrt 5} & \dfrac{2}{3} \\ \dfrac{2}{\sqrt 5} & \dfrac{2}{3 \sqrt 5} & - \dfrac{1}{3} \end{bmatrix}$$